Imagine walking into your introductory level college algebra class. Today you are teaching the fundamental concept of slope. It is likely that the students in this class have had a wide range of exposure to algebraic ideas, from informal algebraic thinking in the early grades to formal work from middle to high school. Intending to ground today’s lesson in students’ prior experiences, you ask: “What do you know about slope?” A small collection of responses from students include:

*1. Rise over run, and I know you need to divide something. That’s my honest answer.*

*2. The first thing I think is y = mx + b as well as change in y over change in x. I also think about point slope formula. I also think about how difficult it is to find the slope of parabolas as you have to find the slopes of tangent lines around the parabola.*

*3. Honestly, I’d freeze. It’s something to do with lines but it’s so confusing.*

*4. Slope is change over time, the rise over run. For graphing purposes I think of it as up 2 over 3, (m = 2/3).*

*5. The distance traveled on the y-axis over the distance traveled on the x-axis. Slope is also the derivative of a function. *

*6. I know that I hated it, but it had to do with graphing points.*

These responses might make you question your next instructional decision. Some remarks describe the slope concept well (response 5). Others contain fragments of the slope concept and might suggest only procedural understandings (1 and 4). Two descriptions appear to reflect knowledge of calculus (2 and 5). Perhaps the greatest dilemma lies in the responses suggesting trauma from previous mathematics experiences (3 and 6). Given the diverse responses, what mathematical and pedagogical decisions might maximize student learning? What does “good teaching” look like in this environment?

Mathematics teachers make a host of decisions during teaching based on students’ ideas like the ones in the vignette. Mathematics teacher educators think deeply about how mathematical knowledge for teaching (MKT) influences those decisions. When we think about teachers using their MKT to plan, we may imagine a lesson designed to teach students mathematics they have not been taught before. We know, however, that in our mathematics curriculum landscape, students have often experienced some formal instruction on the mathematics content at hand. This curricular overlap is particularly salient at systemic transitions, such as between high school and college. How does the use of teachers’ MKT shift when students are routinely bringing prior instructional experiences to the classroom with them?

To frame our goals, consider the teachers who teach early college mathematics courses, many of whom are part-time faculty working across multiple institutions. Historically, these teachers have not had access to professional development opportunities (Blair, Kirkman, & Maxwell, 2013; Gerhard & Burn, 2014; Mesa, 2017). Since many prospective teachers enter their tertiary education experience through these systems, they are likely to be directly impacted by the mathematics teaching they experience there. As the AMTE Standards note, developing well-prepared teachers of mathematics requires a shared vision of mathematics teaching and learning from all stakeholders, including mathematicians and mathematics educators throughout post-secondary systems. Thus, an examination of the MKT and mathematics pedagogical decisions at this early college level is crucial.

### Tacit Assumptions About Teacher Knowledge

MKT has been defined as the unique aspects of content and pedagogical knowledge needed to teach school mathematics beyond other everyday uses of mathematics (Ball, Thames, & Phelps, 2008). Research on MKT has informed teacher preparation, professional development, and curriculum at all levels of instruction (e.g., Ball, Thames, & Phelps, 2008; Gerami, Leckrone, & Mesa, in press; Hill, Rowan, & Ball, 2005; Hill, Sleep, Lewis, & Ball, 2007; Mesa, Celis, & Lande, 2014; Rasmussen & Marrongelle, 2006; Speer, King, & Howell, 2015). Most MKT research to date has embraced two tacit assumptions:

ASSUMPTION 1: Students’ prior knowledge is the primary consideration in designing subsequent mathematical learning.

ASSUMPTION 2: The content is relatively new to the learner.

Consider a concrete instructional situation. A Calculus teacher discussing the chain rule might assume students are familiar with basic differentiation and functions. However, an especially skilled teacher might be deliberate in addressing: (a) function composition as the conceptual heart of the rule, and (b) pattern recognition as a way of connecting new ideas to existing ones. The teacher must balance knowing what students know, building on this knowledge, and anticipating what is challenging for students. The focus on teachers’ skills and decision-making is the very fruit of the aforementioned assumptions, and is central to how MKT is conceptualized.

What if these assumptions do not hold? We argue challenging these assumptions may broaden the utility of MKT to a wider range of instructional situations, including those in which students are bringing unexamined cognitive and socioemotional assets because of curricular overlap. A prime example of that intersection point as illustrated below is early college mathematics, including in community colleges (Mesa, 2017; Mesa, Wladis, & Watkins, 2014), which serve a population that is under-researched.

### Explicit Assumptions in Early College Mathematics

We argue that as a systemic transition with curricular overlap, early college mathematics is a key site for investigating the role of MKT. Specifically, these years are characterized by:

1. developmental (algebra) math sequences;

2. new Pathways intending to align with student majors and/or accelerate students to meet requirements^{[1]};

3. mathematics courses with historically high DFW rates; and,

4. students repeating mathematics classes.

Students who find themselves in one or more of the above situations often share important cognitive and socioemotional characteristics. First, many students self-profess a distaste for mathematics because of a previous trauma or failure. Second, nearly all students have been engaged in formal instruction related to course content. Course placement may have more to do with contextual factors (e.g., placement procedures, major requirements) than previous learning. In sum,* this teaching and learning environment directly challenges the tacit assumptions of MKT research*. Examining effective early college teaching will tell a different narrative that expands and reshapes current perspectives on MKT and teacher decisions.

The above mathematics courses are taught at all institutions of higher education—from 2-year colleges to R1 universities—and we aspire to hear all voices in the conversation on high-quality mathematics teaching. Given our previous assertions, we propose the following two counter-assumptions particularly relevant to early college mathematics:

ASSUMPTION 1: Students' cognitive and socioemotional assets are critical considerations in supporting mathematics learning.

ASSUMPTION 2: The content is familiar to the learner.

Assumption 1 explicitly considers that prior student experiences may induce trauma, posing significant barriers to learning. The student responses to the slope question highlight some of these challenges^{[2]}. Some students may doubt their abilities, hew to a fixed mindset, or delay mathematics requirements as long as possible. These concerns are largely affective in nature. With regard to Assumption 2, in the case of students repeating courses, unfinished learning becomes a thorny question about classroom teaching: What is a teacher to do?

Although these issues are particularly salient in 2-year colleges that disproportionately serve first-generation college students, students from lower socioeconomic backgrounds, students from historically marginalized groups, multilingual learners, and students who place into developmental courses (Mesa, 2017), these instructional experiences occur across a variety of institutions where teachers do not receive adequate professional support. These environments call for reexamining effective mathematics teaching and instructional decision-making. Existing frameworks may need to be amended in order to address the mathematical learning of those students most in need. Specifically, we advocate for articulating how MKT and teacher decision-making change in early college mathematics.

### Looking Ahead

MKT makes important contributions to mathematics teacher education as a way of describing the knowledge, decision-making skills, and resources needed to teach mathematics effectively. Studying MKT in the context of teaching new mathematics content has brought about powerful insights. This focus, however, may bring tacit assumptions that do not hold at some important junctures in the K-16 mathematics curriculum.

When students experience curricular overlap at the secondary and tertiary levels, they bring a wider range of assets to their mathematical work. Developmental mathematics courses are gatekeepers for many, including some prospective K-12 mathematics teachers. Considering the cognitive and socioemotional assets students bring to these courses is critical. While the content may not be novel to students, the deep understandings of that content that teachers hope to develop within students will be. Navigating the assets students bring requires mobilizing different aspects of mathematical knowledge for teaching. Returning to the opening vignette, how might a college algebra teacher make use of the knowledge evident in student responses while being attentive and considerate to the trauma some responses suggest? Broadening the ways in which we think about and study MKT as a construct is critical for better understanding and supporting mathematics teaching at points of curricular overlap.

#### References

Association of Mathematics Teacher Educators. (2017). *Standards for Preparing Teachers of Mathematics*. Available online at amte.net/standards.

Ball, D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for teaching: What makes it special? *Journal of Teacher Education*, *59*(5), 389-407.

Blair, R., Kirkman, E. E., & Maxwell, J. W. (2013). *Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2010 CBMS survey*. Washington, DC: American Mathematical Society.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. *American Educational Research Journal*, *42 *(2), 371-406.

Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts? In F.K. Lester (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 111-155). Charlotte, NC: Information Age Publishing.

Gerami, S., Leckrone, L., & Mesa, V. (to appear). Exploring instructor questioning in community college algebra classrooms and its connections to instructor knowledge and student outcomes. *MathAMATYC Educator*.

Gerhard, G., & Burn, H. E. (2014). Effective engagement strategies for non-tenure-track faculty in precollege mathematics reform in community colleges. *Community College Journal of Research and Practice*, *38*(2-3), 208-217.

Mesa, V. (2017). Mathematics education at U.S. public two-year colleges. In J. Cai (Ed.), *Compendium for Research in Mathematics Education *(pp. 949-967). Reston, VA: NCTM.

Mesa, V., Celis, S., & Lande, E. (2014). Teaching approaches of community college mathematics faculty: Do they relate to classroom practices? *American Educational Research Journal*, *52*, 117-151.

Mesa, V., Wladis, C., & Watkins, L. (2014). Research problems in community college mathematics education: Testing the boundaries of K-12 research. *Journal for Research in Mathematics Education*, *45* (2), 173–192.

Rasmussen, C., & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. *Journal for Research in Mathematics Education, 37* (5), 388-420.

Speer, N. M., King, K. D., & Howell, H. (2015). Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers. *Journal of Mathematics Teacher Education*, *18*, 105-122.

^{[1]} These Pathways go by names such as *New Life* (Mathematical Literacy for College Students and Algebraic Literacy), Dana Center’s *Mathematics Pathways*, and Carnegie Foundation’s *Quantway* and *Statway*. These reforms have received much attention in their widespread implementation at community colleges and universities in the last ten years.

^{[2]} These responses were curated from two university-level mathematics courses—Quantitative Reasoning and Mathematics for Elementary School Teachers.